Student R. is a highly intelligent boy who is motivated to learn math. Moreover, he has an interest in and enjoys the challenge of the course. When a new topic is explained by the teacher in class he listens attentively and understands the algebra of the steps. But when it comes to apply the same steps to a different situation he gets stuck in the middle of the problem, which reveals that he does not have a deep understanding of the logic behind each step.
The problem with R. is that he does not know how to listen to the explanations of Math topics. He did not develop the “Math listening” skill which tells the mathematician to “think about the logical reasons of the steps not the algebraic simplifications”. Student R needs to understand the theoretical, abstract part of the concepts first. Then he has to solve a few more questions with different variations of the same concept applying his knowledge by himself.
When a teacher explains the steps of solving questions the best approach is to listen and ask yourself “WHY are we doing these steps?” Which means “What is the mathematical or logical reason for these steps?” Try to answer by yourself and, if you can’t, ask your teacher. Keep in mind that there is always a chain of reasons behind each step that could be reduced down to basic arithmetic operations, or to very basic definitions.
I like to compare math methods with a tree. Steps are the surface parts of the tree, the branches and leaves,, while the logic and the reasons behind the steps are the roots. The deeper the roots are the stronger the tree is. The deeper you understand the reasons the better you can apply your knowledge when a question changes.
One of the ways to check your understanding is writing summaries in the form of lecture notes. Imagine yourself being a teacher and write short summaries of what you would teach your students.Also, try to help somebody understand a topic if you feel you have learned it well.